x^2 + y + 7 = 7 20x + 100 - y = 0 The solution to the given system...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{x^2 + y + 7 = 7}\)
  • \(\mathrm{20x + 100 - y = 0}\)
• Need to find: value of x

2. INFER the solution strategy

• Since we have two equations and two unknowns, use substitution method
• The second equation is linear in y, making it easier to solve for y first
• Strategy: solve second equation for y, then substitute into first equation

3. SIMPLIFY each equation separately

• First equation: \(\mathrm{x^2 + y + 7 = 7}\)
  \(\mathrm{x^2 + y = 0}\)
  \(\mathrm{y = -x^2}\)
• Second equation: \(\mathrm{20x + 100 - y = 0}\)
  \(\mathrm{y = 20x + 100}\)

4. INFER the next step and set up the substitution

• Since both expressions equal y, set them equal to each other
• \(\mathrm{-x^2 = 20x + 100}\)

5. SIMPLIFY to standard quadratic form

• \(\mathrm{-x^2 = 20x + 100}\)
• \(\mathrm{-x^2 - 20x - 100 = 0}\)
• Multiply by -1: \(\mathrm{x^2 + 20x + 100 = 0}\)

6. INFER the factoring approach

• Recognize this as a perfect square trinomial
• Pattern: \(\mathrm{a^2 + 2ab + b^2 = (a + b)^2}\)
• Here: \(\mathrm{x^2 + 20x + 100 = (x + 10)^2}\)

7. SIMPLIFY to find the solution

• \(\mathrm{(x + 10)^2 = 0}\)
• Taking square root: \(\mathrm{x + 10 = 0}\)
• Therefore: \(\mathrm{x = -10}\)

Answer: -10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make sign errors when rearranging \(\mathrm{-x^2 = 20x + 100}\) to standard form.

They might incorrectly write \(\mathrm{x^2 - 20x - 100 = 0}\) instead of \(\mathrm{x^2 + 20x + 100 = 0}\), missing the negative sign on the 20x term. This leads to a completely different quadratic that doesn't factor nicely, causing them to get stuck or use the quadratic formula incorrectly.

This leads to confusion and guessing among the available choices.

Second Most Common Error:

Missing conceptual knowledge of perfect square trinomials: Students may correctly reach \(\mathrm{x^2 + 20x + 100 = 0}\) but fail to recognize the perfect square pattern.

Instead of seeing \(\mathrm{(x + 10)^2}\), they might try to factor using trial and error or jump straight to the quadratic formula, potentially making calculation errors that lead them away from the clean answer of \(\mathrm{x = -10}\).

This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests both systematic algebraic manipulation and pattern recognition. The key insight is recognizing that the resulting quadratic is a perfect square, which makes the solution much cleaner than students initially expect.